     James Joseph Sylvester
Collected Mathematical Papers, Vol. I
     New work: book pp. 595–608
                                          59.
 On Mr Cayley’s Impromptu Demonstration of the Rule for
   Determining at Sight the Degree of any Symmetrical
Function of the Roots of an Equation Expressed in Terms of
                     the Coefficients
                [Philosophical Magazine, V. (1853), pp. 199–202]
                                                                                          p. 595
   For a considerable time past, among the few cultivators of the higher algebra,
a proposition relative to the theory of the symmetrical functions of the roots
of an equation has been in private circulation, which, to say nothing of the
important applications of which it has been found susceptible to the calculus
of forms, merits (by reason of its extreme simplicity), although, strange to say,
it has, I believe, not yet obtained, a place in elementary treatises on algebra.
The proposition alluded to I have reason to think first came to be observed in
connexion with my well-known formulae for Sturm’s auxiliary functions in terms
of the roots given in this Magazine. The theorem is briefly as follows. If a, b, c,
&c. be the roots of an equation
                          xn + p1 xn−1 + p2 xn−2 + &c. = 0,
any symmetric function such as Σaα bβ cγ . . ., where α, β, γ . . . are positive integers
arranged according to the order of their magnitudes in a descending (or, to
speak more strictly, non-ascending) order, when expressed as a function of the
coefficients, will be made up of terms of the form pθ11 pθ22 pθ33 . . . pθkk , such that
θ1 + θ2 + θ3 + . . . + θk will be equal to α for some terms, but will for no term
exceed α; α being, as above described, that one of the indices α, β, γ . . . which is
not less than any of the others.
   I had prepared, and indeed despatched, a somewhat elaborate proof of this
theorem for the Cambridge and Dublin Mathematical Journal; but on proceeding
to explain my method to Mr Cayley, elicited from that sagacious analyst the
following excellent impromptu, which I think too valuable to be lost; and as it is
now a twelvemonth or two since our conversation on the subject took place, and
the author has not cared to put it on record, I feel                                      p. 596
   myself under an obligation so to do, the more so as it entirely supersedes
the comparatively inelegant demonstration of my own which I had previously
intended to publish.
   The method rests essentially on the following well-known theorem given by
Euler relative to the partition of numbers; to wit, that the number of ways of
breaking up a number n into parts is the same, whether we impose the condition
that the number of parts in any partitionment shall not exceed m, or that
the magnitude of any one of the parts shall not exceed m. Of this rule more
hereafter—for the present to its application to the matter in hand.

                                            2
   Since a, b, c . . . are the roots of xn + p1 xn−1 + . . ., we have
                             p1 = a + b + c + . . . ,
                             p2 = ab + ac + bc + . . . ,
                             p3 = abc + abd + acd + . . . ,
                                ············
                                ············ .
Let α + β + γ + . . . = n, none of the quantities α, β, γ . . . being greater than m,
but α, β, γ . . . being otherwise arbitrary and capable of becoming equal to any
extent inter se. Also let λ + µ + ν + . . . = n, the number of quantities λ, µ, ν,
&c. being never greater than m, but the quantities themselves being otherwise
arbitrary, and being capable of becoming equal to any extent inter se. By Euler’s
rule the number of systems α, β, γ . . . is the same as of the systems λ, µ, ν . . ., say
P for each. For any system λ, µ, ν . . ., we shall have pλ pµ pν . . ., by virtue of the
equations above written, expressible as the sum of terms of the form Σaα bβ cγ . . .;
it may easily be made ostensible, that all the combinations of α, β, γ . . . subject
to the above prescribed conditions must come into evidence by giving λ, µ, ν . . .
all the variations of which they admit; but this is also immediately obvious
indirectly from the consideration, that were it otherwise, linear relations would
subsist between the different values of pλ pµ pν . . ., which is obviously absurd.
Hence, then, we shall be able to express the P quantities of the form pλ pµ . . .
by means of linear functions of the P quantities Σaα bβ cγ . . .; and conversely, by
solving the linear equations thus arising, the P quantities Σaα bβ cγ . . . may be
expressed in terms of the quantities pλ pµ . . .; consequently Σaα bβ cγ . . ., where m
is greater or not less than any of the quantities β, γ . . ., will be expressible by
means of combinations pλ pµ . . ., where the number of coefficients pλ pµ . . . (any
number of which may become identical) is for some of the combinations as great
as, but for none of the combinations greater than m, as was to be proved. It
will of course be seen that, for the purposes of the demonstration above given, it
would have been sufficient                                                                p. 597
   to have been able to assume that the number of partitions, when the greatest
part is not allowed to exceed m, is not greater than the number of partitions when
the number of parts in any one partitionment does not exceed m. The equality
of these two numbers would then evince itself in the course of the demonstration
as a consequence of this assumption.
   A word now as to Euler’s beautiful law upon which the above demonstration
is based.
   A corollary from it, obtained by subtracting the equation which it gives when
the limiting number is taken (m − 1) from the equation which it gives when the
limiting number is m, will be the following proposition. The number of modes
of partitioning n into m parts is equal to the number of modes of partitioning
n into parts, one of which is always m, and the others m or less than m. This

                                            3
proposition was mentioned to me by Mr N. M. Ferrers1 , whose demonstration
of it (probably not different from that of Euler’s for the other proposition, of
which it may be viewed as a corollary) is so simple and instructive, that I am
sure every logician will be delighted to meet with it here or elsewhere. It affords
a most admirable example of that rather uncommon kind of reasoning whereby
two abstract integers are proved to be equal indirectly, by showing that neither
can be greater than the other.
   If there be a group of A’s and a group of B’s, and every A can be shown to
produce a B, and every B can be shown to produce an A, no matter whether
the A producing a B is the same as, or different from, the A produced by that
B, it is obvious that the number of A’s cannot exceed that of the B’s, nor of the
B’s that of the A’s, and the two numbers will therefore be equal.
   Take any such grouping as 3, 3, 2, 1, say A. This may be written as

                                       1, 1, 1
                                       1, 1, 1
                                       1, 1,
                                       1,

and by reading off the columns as lines, may be transformed into the group

                                    1, 1, 1, 1
                                    1, 1, 1
                                    1, 1

that is 4, 3, 2, say B.                                                            p. 598
   In A the number of parts is 4. In B the greatest part is 4; the others might
be (although they happen not in this particular instance to be) 4, but cannot
be greater than 4. And so every A in which the number of parts is 4 will give
rise to a B in which 4 is one of the parts, and every other part is 4 or less, and
evidently (although, as above remarked, this is immaterial to the demonstration)
every such B gives reciprocally the same A from which it is itself derived; hence
the number of A’s and B’s is equal. This is the theorem which, for the sake of
distinction, I have called the Corollary to Euler’s. Euler’s own is proved by the
same diagram; for if we define A as a grouping where the number of parts does
not exceed 4, we get a definition of B as a grouping where the greatest part does
not exceed 4, and so in general. We see that this theorem may be varied also by
affirming that the number of ways in which n may be broken up, so that there
shall never be less than m parts, is the same as the number of ways in which it
may be broken up into parts, the greatest of which in any one way is not less
than m. So, again, a similar diagram makes it apparent, that if we break up each
of i numbers into parts so that the sum of the greatest parts shall not exceed
  1
    I learn from Mr Ferrers that this theorem was brought under his cognizance through a
Cambridge examination paper set by Mr Adams of Neptune notability.


                                           4
(or be less than) m, the number of ways in which this can be done will be the
same as the number of ways in which these i numbers can be simultaneously
partitioned so that the total number of parts in any simultaneous partitionment
shall never exceed (or never be less than) m; and doubtless an extensive range of
analogous general theorems relative to the partitioning of numbers may be struck
out by aid of the same diagram, by no means easily demonstrable unless this
simple mode of conversion happen to be thought of, but in that event becoming
intuitively apparent. This mode of conversion is precisely that (only applied to a
more general state of things) whereby, in elementary arithmetic, it is established
that m times n is the same as n times m. A consideration of the process by
which the mind satisfies itself of the universality of this law, has been always
sufficient to convince me of the absurdity of ascribing to an inductive process
the capacity of the human mind for forming general ideas concerning necessary
relations.




                                        5
                                              60.
A Proof that all the Invariants2 to a Cubic Ternary Form
are Rational Functions of Aronhold’s Invariants and of a
     Cognate Theorem for Biquadratic Binary Forms
            [Philosophical Magazine, V. (1853), pp. 299–303, 367–372]
                                                                                                     p. 599
   Although contrary to the order of exposition indicated in the title to this
paper, I shall, as the simpler case, begin with establishing the theorem for a
biquadratic form, say F in x, y. Let

                       F = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ,
                        s = ae − 4bd + 3c2 ,
                        t = ace − ad2 − c3 − b2 e + 2bcd,

s and t are the two well-known invariants of F . I propose to prove that there
can exist no other invariants to F except such as are explicit rational functions
of s and t.
   Let F , by means of the substitution of f x + gy for x, and f ′ x + g ′ y for y,
be made to take the form f1 = x4 + y 4 + 6mx2 y 2 . Then by the characteristic
property of invariants, if I(a, b, c, d, e) be any invariant to F of the degree q, we
must have
                   I(1, 0, m, 0, 1) = (f g ′ − f ′ g)2q I(a, b, c, d, e);
and it will be sufficient to prove that I(1, 0, m, 0, 1), or say more simply I(m),
can only have the two radically distinct forms corresponding to s and t, that is

                          (s) = 1 − 3m2       and     (t) = m − m3 ,

any other admissible form of I being a rational explicit function of these two. p. 600
   It may be shown3 that the parameter m in f1 will have six different  √ values
and no more. In the first place, if we write ix for x in f1 (i meaning −1), it
is obvious that m becomes −m. Again, let x + iy and x − iy be substituted in
place of x and y respectively; then calling (f ) the value assumed by f1 , when
   2
     A Constant in analysis is any quantity which in its own nature, or by the explicit conditions
to which it is subjected, is incapable of change. An Invariant is an expression apparently liable
to change, but which, owing to certain compensations in the modifying tendencies impressed
upon it, remains as a whole unaltered. The former may be compared to a fixed point or system
in mechanics; the latter to a point or system free to move, but kept at rest under the combined
operation of contending forces.
   3
     See Addendum [p. 607 below].




                                                6
this substitution is made,

                    (f ) = (x + iy)4 + (x − iy)4 + 6m(x2 + y 2 )2
                       = (2 + 6m)(x4 + y 4 ) + (−12 + 12m)x2 y 2
                                                −1 + m 2 2
                                                          
                       = (2 + 6m) x4 + y 4 + 6          x y .
                                                 1 + 3m
Hence if we write
                            1               i
                               1/4
                                   x+             y         for x,
                       (2 + 6m)       (2 + 6m)1/4

and
                             1               i
                                    x−             y        for y,
                        (2 + 6m)1/4    (2 + 6m)1/4
and call what f1 becomes after these substitutions f2 ,

                             f2 = x4 + y 4 + 6γ(m)x2 y 2 ,
              −1 + m
γ(m) denoting         .
              1 + 3m
  In like manner, by writing in f2
                           1                 i
                               1/4
                                   x+                y          for x,
                    {2 + 6γ(m)}       {2 + 6γ(m)}1/4

and
                           1                 i
                                   x−                y          for y,
                    {2 + 6γ(m)}1/4    {2 + 6γ(m)}1/4
we obtain
                             f3 = x4 + y 4 + 6γ 2 (m)x2 y 2 ,
where
                              −1 + m
                           −1 +
                γ 2 (m) =     1 + 3m = −2 − 2m = −1 − m ;
                              −1 + m   −2 + 6m   −1 + 3m
                          1+3
                              1 + 3m
γ(m) is a periodic function of m of the third order, for we find

                                        −(1 + 3m) − (−1 + m)
              γ 3 (m) = γ 2 {γ(m)} =                          = m.
                                        −(1 + 3m) + 3(−1 + m)

It will of course be observed, also, that

                  γ 2 (m) = −γ(−m) and           γ(m) = −γ 2 (−m).
                                                                         p. 601




                                            7
  Hence

    (−γ)(−γ)(m) = −γ 3 (−m) = m,                 (−γ 2 )(−γ 2 )(m) = −γ 3 (−m) = m.

So that, in fact, the six values of the parameter are

                              m,  γ(m),   γ 2 (m),
                              −m, −γ(m), −γ 2 (m),

forming two cycles, having the remarkable property that the terms in the same
cycle are periodic functions of the third order of one another, and each term in
one cycle is a periodic function of the second order of every term in the other
cycle.
   The modulus of substitution for passing from f1 to f2 , that is the square of
the determinant
                                 1              i
                           (2 + 6m) 1/4   (2 + 6m)1/4
                                 1             −i       ,
                           (2 + 6m)1/4 (2 + 6m)1/4
is
                             (−2i)2             −2
                                     , or           .
                             2 + 6m          1 + 3m
So that if I(m) be the value of any         of the degree q, corresponding to
                                   invariant
                                    m−1
the form f1 , and consequently I              the same for f2 , we must have
                                   1 + 3m
                                                q
                                       1 + 3m                m−1
                                                                    
                        I(m) =                       I              .
                                         −2                  1 + 3m
In like manner, by means of f3 it may be shown that we must have the further
equation
                                1 − 3m q     m+1
                                                 
                      I(m) =              I           .
                                  −2         1 − 3m
These equations are easily verified for the values of (s) and (t).
  Thus
                                                (                           2 )
                            (1 + 3m)2                m−1
                                                             
            (s) = 1 + 3m2 =                      1+3
                                4                    3m + 1
                               (                         2 )
                  (1 − 3m)2          m+1
                                         
                =                1+3                             ,
                      4              1 − 3m
                                                 (                                 3 )
                             (1 + 3m)3               m−1                   m−1
                                                                      
                          3
             (t) = m − m = −                                −
                                 8                   3m + 1               3m + 1
                                   (                                 3 )
                   (1 − 3m)3           m+1               m+1
                                                     
                =−                            −                             ;
                       8               1 − 3m            1 − 3m

                                             8
                                                                                        p. 602
   and it is moreover obvious, that the values of (s) and (t) might have been
found à priori by means of these functional equations.
   The essential point of inference for my present purpose from the equations
above, which are of the form
                                     m−1                      m+1
                                                                
                 I(m) = H × I                     =K ×I              ,
                                     3m + 1                   1 − 3m

is this, that if I(m) contain any power of m, say mi , it must also contain (m − 1)i
and (m + 1)i ; in a word, (m3 − m)i , which, by the way, it may be noticed, is (t)i .
Now, if possible, let there be any invariant Iq (m) of the qth degree in m which
is not a rational function of (s) and (t). If we make

                                     2x + 3y = q,

as many integer solutions as exist of this equation (in which zero values of x and
y are admissible), so many functions of the form (s)x (t)y may be formed of the
degree q in m, and all of them of course invariantive functions.
    As regards the general nature of any invariantive function in m, since the
change of x into −x in x4 + y 4 + 6mx2 y 2 introduces no change into the invariant
if q be even, but changes the sign if q be odd, it follows that Iq (m) is of the form
ϕ(m2 ) when q is even, and of the form mϕ(m2 ) when q is odd.
    Let µ be the number of solutions of the equation in integers above written.
Then, by linearly combining all the different values of (s)x (t)y with Iq (m), it is
obvious that we may form a new invariant, say Iq′ , in which the µ first occurring
powers of m will be wanting, that is in which the indices 0, 2, 4 . . . (2µ − 2) will
be wanting when q is even, and 1, 3, 5 . . . (2µ − 1) when q is odd. Hence in the
former case the new invariant will contain m2µ , and in the latter case m2µ+1 ; and
therefore, by virtue of what has been shown already, Iq′ will contain (m3 − m)2µ
in the one case and (m3 − m)2µ+1 in the other.
    Firstly, let q = 6i, or 6i+2, or 6i+4; then µ = i+1; and therefore (m3 −m)2i+2 ,
which is of the degree 6i + 6 in m, is contained as a factor in I which is of the
degree q only, a quantity less than 6i + 6, which is absurd.
    Again, secondly, let q = 6i + 1, then µ = i; and (m3 − m)2µ+1 is of the degree
6i + 3 in m, and is contained as a factor in I, which is of the degree 6i + 1, which
is again absurd.
    Finally, if q = 6i + 5, or 6i + 3, µ = i + 1; and the factor (m3 − m)2µ+1 is of
the degree 6i + 9, that is, in each case, greater than q, which is absurd, and thus
the theorem is completely demonstrated.
    It may for a moment be objected, that we have been dealing only with a
particular form x4 + 6mx2 y 2 + y 4 , instead of the general form

                      ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ;

                                          9
                                                                                                    p. 603
   but the latter is always reducible to the former by means of a definite linear
substitution; and if we call the modulus of the substitution, that is the square of
the determinant formed by the coefficients of substitution, M , to every general
invariant Iq of the qth degree, to the latter corresponds a partial form (Iq ) of
invariant to the former, such that
                                               1
                                        Iq =      (Iq );
                                               Mq
and consequently, since every (I) is a rational function of (s) and (t), so must
every I be the same of s and t; unless, indeed, it were possible to have
                                                1
                                        Iq =        (Iq′ ),
                                               M q′
q ′ being different from and greater than q: but if this were the case, since
       1
Iq = q (Iq ), a power of M the modulus would necessarily be an invariant; but
      M
in passing from x4 + y 4 + 6mx2 y 2 to x4 + y 4 + 6γ(m)x2 y 2 , 1 + 3m becomes the
modulus, which we know is not an invariant. Hence the proposition is completely
established for the case of the biquadratic function (x, y)44 .
    Now let us proceed to Aronhold’s famous S and T , the invariants to the
general cubic function (x, y, z)3 , forms equally dear to the analyst and geometer.
(Vide Mr Salmon’s Higher Plane Curves passim.)
    The method will be precisely the same as that applied to s and t5 .
    We commence with the canonical form

                                  x3 + y 3 + z 3 + 6mxyz.

On substituting x + y + z, x + ρy + ρ2 z, x + ρ2 y + ρz for x, y, z, where ρ is the
cube root of unity, the above quantity takes the form

                         (3 + 6m){x3 + y 3 + z 3 + 6β(m)xyz},

where
                                        18 − 18m    1−m
                             β(m) =               =        ,
                                        6(3 + 6m)   1 + 2m
a periodic function in m of the second order only, for
                                         1 + 2m − 1 + m
                            β 2 (m) =                   = m.
                                        1 + 2m + 2 − 2m
                                                                                                    p. 604
   4
     I have made a tacit assumption throughout the foregoing demonstration (which is, however,
capable of an easy proof), namely that if any fractional function of the coefficients of any form
be invariantive, the numerator and denominator must be separately invariants.
   5
     The S is Mr Cayley’s property, the T belongs to Professor Boole, having been by him
imparted, in the infancy of the theory, to Mr Cayley, by whom it was first given to the world,
at least in its character as an Invariant.


                                                10
   But if we write for x in the original form ρx, it becomes
                             x3 + y 3 + z 3 + 6ρmxyz;
and if for x we write ρ2 x, it becomes
                             x3 + y 3 + z 3 + 6ρ2 mxyz.
Hence we can by linear substitutions obtain from x3 + y 3 + z 3 + 6mxyz the three
additional forms
                          x3 + y 3 + z 3 + 6β(m)xyz,
                            x3 + y 3 + z 3 + 6γ(m)xyz,
                            x3 + y 3 + z 3 + 6δ(m)xyz,
where
                        1−m                    1 − ρm     ρ2 − m
              β(m) =           ,   γ(m) = ρ2           =         ,
                        1 + 2m                 1 + 2ρm   1 + 2ρm
                                  1 − ρ2 m      ρ−m
                        δ(m) = ρ        2
                                            =           .
                                 1 + 2ρ m     1 + 2ρ2 m
In all, there will be twelve values of m forming three remarkable compound
cycles,
                        m,     β(m),      γ(m),    δ(m),
                       ρm, ρβ(m), ργ(m), ρδ(m),
                       ρ2 m, ρ2 β(m), ρ2 γ(m), ρ2 δ(m).
It would be beside my present object to seek to develope fully the functional
relations in which the several terms of these cycles stand to one another: the
interesting relations
                         β 2 (m) = γ 2 (m) = δ 2 (m) = m,
                             βγ(m) = γβ(m) = δ(m),
                             γδ(m) = δγ(m) = β(m),
                             δβ(m) = βδ(m) = γ(m),
have been already6 stated by me in another place (Cambridge and Dublin
Mathematical Journal, March 18517 ).
   The (S) of the canonical form corresponding to the S of the general form is
m−m4 ; and the (T ) corresponding to the T of the general form is 1−20m3 −8m6 .
(See my Calculus of Forms8 , Cambridge and Dublin Mathematical Journal,
February 1852.) It is my object to show that any other invariant (I) to the
canonical form must be a rational function of S and T .
   In the first place, I observe that every invariant to any function of an odd
degree i of any odd number p of variables must be of even dimensions; for if the
degree of the dimensions be q, and D the determinant of the                      p. 605
  6
    p. 192 above.
  7
   Vide Addendum [p. 607 below].
  8
    p. 311 above.


                                         11
   coefficients of substitution, the invariant to the transform becomes the original
                                                   iq
invariant affected with a factor Diq/p , where        must be an even integer, since
                                                   p
otherwise the sign of this multiplier would be equivocal and indeterminable;
hence when i and p are both odd, q must be even. Thus, then, I(m) in the case
before us must be an even-degreed function of m. Moreover, since the change of
x into ρx converts m into ρm, and Iq (m) into ρq Iq (m), for D becomes ρ when
x, y, z become ρx, y, z, Iq (m) must be of the form ϕ(m3 ), m2 ϕ(m3 ), mϕ(m3 ),
according as the index q is of the form 6i, 6i + 2, 6i + 4.
   By precisely the same reasoning as was applied to the preceding case of (s)
and (t), we see that any invariant of m which contains mc must also contain
(1 − m)c , (1 − ρm)c , (1 − ρ2 m)c , that is must contain (m − m4 )c , which in fact
is (S)c . If, now, we consider any invariant of the qth degree in m, I(m), and
suppose it to be other than a rational function of (S) and (T ), and if we take µ
to denote the number of the solutions of

                                      4x + 6y = q,

it will follow that we may form an invariant I ′ (m), which, when q is of the form
12i or 12i + 6, will contain m, and consequently (m − m4 )3µ+2 as a factor; and in
like manner when q is of the form 12i + 2 or 12i + 8, will contain (m − m4 )3µ+2 as
a factor; and when q is of the form 12i + 4 or 12i + 10 will contain (m − m4 )3µ+1
as a factor. Now when
                                   q = 12i,     µ = i + 1,
                               q = 12i + 6,     µ = i + 1;

when
                               q = 12i + 2,     µ = i,
                               q = 12i + 8,     µ = i + 1;
when
                               q = 12i + 10,    µ = i + 1,
                                q = 12i + 4,    µ = i + 1.
Hence the factors dividing Iq in these several cases will be of the respective
degrees

       12i + 12, ; 12i + 12;       12i + 8, 12i + 12;        12i + 16, 12i + 16;

corresponding to q, being of the several values

            12i, 12i + 6;        12i + 2, 12i + 8;      12i + 10, 12i + 4;

which is clearly impossible. This proves the theorem in question (the passage
being made from the canonical to the general form, as in the former part of
this investigation), to wit, that S and T form what I have elsewhere termed

                                           12
a fundamental scale of invariants to the cubic ternary form, entering as the
exclusive ingredients into every other invariant that can be derived from such
form.                                                                                 p. 606
   A word of warning is necessary before I lay down my pen: that there can
be only two algebraically independent invariants to (x, y)4 or (x, y, z)3 , is an
immediate consequence of the canonical form of each having but one parameter;
so in general there can be at most but (n − 2) absolutely independent invariants
of (x, y)n ; but the point established in the preceding investigation goes to show
that there can exist no other invariants than such as are rational functions of s
and t in the one case, and S and T in the other. I shall take some other occasion
to establish a similar conclusion for the forms (x, y)5 and (x, y)6 .
   I have shown that there exist three invariants to the one of the degrees 4, 8, 12,
and four to the other of the degrees 2, 4, 6, 10; and I shall demonstrate that any
other invariant to either form must be a rational function of those above stated.
For the cubic form (x, y)3 we know that there is but one invariant, namely its
discriminant. Thus, then, for n = 3, n = 4, n = 5, n = 6 the number of absolutely
independent invariants is n−2, and the number of linearly independent invariants
is no greater. But this result is by no means generally true. It may be proved by
means of a great law of reciprocity9 which I myself originated, but unfortunately
threw aside, and which M. Hermite has since demonstrated, that there are more
than five linearly independent invariants to (x, y)8 , and more than ten, in fact
twelve at least, to (x, y)12 ; that is to say, it is impossible in the latter case to
find ten of which all the rest shall be rational functions, although an algebraical
equation connects any 11. So, again, if we take a system of two cubic equations,
there are only five absolutely independent invariants; but there are not less than
seven linearly independent fundamental invariants,                                    p. 607

   9
     The theorem of reciprocity alluded to in the text is the following:—If to any function (x, y)n
there exists an invariant of the order m in the coefficients, then to (x, y)m there exists an
invariant of the order n in the coefficients; or more generally, which is M. Hermite’s addition, if
to any system of functions (x, y)n1 , (x, y)n2 . . . (x, y)ni there exists an invariant of the several
dimensions m1 , m2 . . . mi in the respective sets of coefficients, then conversely to a system
(x, y)m1 , (x, y)m2 . . . (x, y)mi there exists an invariant of the dimensions n1 , n2 . . . ni in the
respective sets of coefficients.
   I had previously shown in this Magazine [p. 279 above], that Mr Cayley’s formulae for finding
the number of biquadratic invariants to any function (x, y)n , given in that remarkable paper of
his on linear transformations [Cayley’s Collected Papers, Vol. I., p. 95], where first dawned upon
the world the clear and full-formed idea of invariants (the most original and important infused
into analysis since the discovery of fluxions), could be expressed by means of the number of
solutions of the equation in integers 2x + 3y = n, the square of the quadratic invariant (which
only exists for even values of n) counting for one in the fundamental biquadratic scale; this
is of course a direct consequence, through the law of reciprocity, of the fundamental scale to
(x, y)4 consisting of a quadratic and a cubic invariant. My discovery of the fundamental scale of
invariants to (x, y)5 and (x, y)6 now enables us, through the same law of reciprocity, to express
the number of distinct Quintic and Sextic invariants to (x, y)n , namely as being the number of
integer solutions of x + 2y + 3z = n4 in the one case, and of x + 2y + 3z + 5t = n2 in the other.



                                                 13
   of which any other invariant must be a rational function. In fact, if we take
for our two cubics
                       U = ax3 + 3bx2 y + 3cxy 2 + dy 3 ,
                        V = αx3 + 3βx2 y + 3γxy 2 + δy 3 ,
the five coefficients of the powers of λ, in the discriminant of U + λV , each
of which is of four dimensions in the two sets of coefficients combined, are all
invariants of the system; but there will be besides two more, one of which is a
Combinant of six dimensions, being the resultant of U and V ; the other is a
Combinant of two dimensions only, namely aδ − 3bγ + 3cβ − dα. These seven
together form the fundamental constituent scale.
   The two last-mentioned may be expressed algebraically (by the introduction of
square roots) as functions of the other five, but of course not as rational functions
of the same. My attention was more particularly called to the search of a proof
of the completeness of the Aronholdian system of invariants, by an inquiry as
to the possibility of rigidly demonstrating that there could exist no others not
made up of these, addressed to me in the spring of last year by one of the most
gifted geometers of this or any other country. A morning or two after the inquiry
reached me, in a walk before breakfast by the side of the ornamental water in
St James’s Park (a time and place by no means, according to my experience,
unfavourable to the inspirations of the analytic muse), I had the satisfaction of
falling upon the rather piquant demonstration above given, which essentially
rests upon a principle, requiring no harder exercise of faith than the concession
of the impossibility of a greater being contained in or proceeding out of a less.

                                   Addendum.

On the nature of the three Cycles of four terms each which contain the twelve
 values of the parameter to the canonical form of a cubic function of three
                                  variables.

   The equations given in the text [p. 604 above] show that each term in any one
cycle is a periodic function of the second order of each other term in the same
cycle. Moreover, it may be shown that each term in any one cycle is a periodic
function of the third order of every term in either of the other two cycles; a sort
of relation between the cycles taken per se, and with one another, precisely the
inverse of what obtains (as already shown) for the two cycles of three terms
containing the six values of the parameter to the biquadratic function of two
variables. For as regards that case, it was shown                                   p. 608
   in the first part of this paper that the terms in the same cycle are periodic
functions of the third order of one another, and of the second order of each of
those not in the same cycle with themselves.


                                         14
  If we make
                      1−m                     ρ2 − m                 ρ−m
       m = A,                = B,                    = C,                    = D,
                      1 + 2m                 1 + 2ρm               1 + 2ρ2 m

                ρA = A′ ,      ρB = B ′ ,         ρC = C ′ ,      ρD = D′ ,
           ρ2 A = A′′ ,      ρ2 B = B ′′ ,        ρ2 C = C ′′ ,     ρ2 D = D′′ .
The following table will exhibit all the ternary periods that can be formed
between the terms of the several cycles:—

          (1)    AB ′ D′′ , (4) BA′ C ′′ ,    (7) CA′ D′′ , (10)         DA′ B ′′ ,
          (2)    AC ′ B ′′ , (5) BC ′ D′′ , (8) CB ′ A′′ ,        (11)   DB ′ C ′′ ,
          (3)    AD′ C ′′ , (6) BD′ A′′ , (9)         CD′ B ′′ , (12)    DC ′ A′′ .

For instance, as an example of the meaning of the table, take line (8), namely
CB ′ A′′ . This indicates that A′′ is formed from B ′ and C from A′′ in the same
way as B ′ from C, and of course A′′ from C in the same way as C from B ′ and
B ′ from A′′ , &c. By means of this table it will easily be seen that a term in each
of two cycles being given, the term in the third which forms with the given two
a ternary period may immediately be assigned.
   The remarks which I have to add on the nature of the equations for finding
the parameter m, as well for (x, y)4 as for (x, y, z)3 , will be given hereafter.




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